Journal of Operator Theory

Volume 41, Issue 1, Winter 1999 pp. 3-22.

Completely multi-positive linear maps and representations on Hilbert $C^*$-modules

Authors: Jaeseong Heo
Author institution: Department of Mathematics, Seoul National University, Seoul 151-742, Korea

Summary: We introduce the notion of (completely) multi-positive linear maps between $C^*$-algebras, and show that a completely multi-positive linear map induces a representation of a $C^*$-algebra on Hilbert $C^*$-modules. This generalizes the Stinespring's representation and the representations constructed by Paschke and Kaplan as well as the GNS representation. We also construct the covariant representations on Hilbert $C^*$-modules for covariant completely positive linear maps. Using representations of $C^*$-algebras on Hilbert $C^*$-modules associated with completely multi-positive linear maps we establish another approach about representations associated with completely bounded linear maps.

Keywords: Completely multi-positive maps, covariant multi-positive maps, covariant representations, Hilbert $C^*$-module representations, injective $C^*$-algebras

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